An Efficient Temporal Two-Mesh Compact ADI Method for Nonlinear Schrödinger Equations with Error Analysis

In this article, we present an efficient numerical strategy for the two-dimensional nonlinear Schrödinger equation, focusing on its development and analysis. Our approach begins with proposing a nonlinear, energy-conservative, fourth-order, compact, alternating-direction, implicit (ADI) scheme. To boost efficiency when solving the associated nonlinear system, we then implement this scheme using a temporal two-mesh (TTM) algorithm. Under discretization with coarse time step ${\tau }_{{}_C}$, fine time step ${\tau }_{{}_F}$, and spatial mesh size h, the numerical scheme exhibits a convergence rate of order $\mathcal{O}({\tau }_{{}_C}^4+{\tau }_{{}_F}^2+h^4)$ in both the discrete $L_2$-norm and $H_1$-norm. To facilitate the convergence analysis under fine time discretization, we propose a novel technique along with several supporting lemmas that enable the estimation of the discrete $L_4$-norm error term over the temporal coarse mesh. Numerical experiments are then performed to validate the theoretical results and demonstrate the effectiveness of the proposed algorithm. The numerical results show that the new algorithm produces highly accurate results and preserves the conservation laws of mass and energy. Compared with the fully nonlinear compact ADI scheme, it reduces computational time while maintaining accuracy.